Subgroups — Q₃₂×Dic₂₃₉ #44486407

Order 30592.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₂₃₉	C₂	1	2	yes	maximal
2	#29504598	C₂	1	1	yes	maximal, characteristic
3	#29504576	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₂₃₉	—	2	2	no	—
7	C₂×C₂₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#21439894	C₂²	1	2	yes	—
9	#21439884	—	2	2	no	—
10	C₈×Dic₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₂₃₉	C₂²	1	2	yes	—
12	Q₃₂×C₂₃₉	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₂₃₉⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₂₃₉	—	2	2	no	—
16	C₄⋊Dic₂₃₉	—	4	2	no	—
17	C₄×Dic₂₃₉	—	4	2	no	—
18	C₄×Dic₂₃₉	D₄	1	1	yes	characteristic
19	C₂₃₉⋊C₁₆	—	2	1	no	—
20	C₂×Q₈×C₂₃₉	—	2	2	no	—
21	Q₁₆×C₂₃₉	—	2	2	no	—
22	Q₁₆×C₂₃₉	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
24	C₁₆×C₂₃₉	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₃₉	—	4	2	no	—
26	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
27	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
28	C₂₃₉⋊C₈	—	2	1	no	—
29	Q₈×C₂₃₉	—	2	4	no	—
30	Q₈×C₂₃₉	—	4	2	no	—
31	C₂×C₄×C₂₃₉	—	4	2	no	—
32	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₃₉	#41	1	1	yes	characteristic
35	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₃₉	—	4	4	no	—
37	C₄×C₂₃₉	C₄×D₄	1	1	yes	characteristic
38	C₄×C₂₃₉	#135	1	1	yes	characteristic
39	Dic₂₃₉	—	2	1	no	—
40	Dic₂₃₉	—	8	2	no	—
41	Dic₂₃₉	Q₃₂	1	2	yes	direct product factor
42	C₂×C₂₃₉	C₄×D₈	1	1	yes	characteristic
43	C₂×C₂₃₉	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₂₃₉	#508	1	1	yes	characteristic
45	C₂₃₉	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₃₂	—	239	1	no	maximal, Sylow
47	C₂×Q₃₂	D₂₃₉	1	1	yes	characteristic
48	C₄×Q₁₆	—	239	2	no	—
49	#366	—	239	1	no	—
50	#358	—	239	2	no	—
51	C₄×C₁₆	—	239	1	no	—
52	C₂×Q₁₆	C₂×D₂₃₉	1	2	yes	—
53	C₄×Q₈	—	478	2	no	—
54	Q₃₂	Dic₂₃₉	1	4	yes	direct product factor
55	C₂×C₁₆	—	239	1	no	—
56	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
57	#107	—	239	2	no	—
58	#103	—	478	2	no	—
59	C₄×C₈	—	239	1	no	—
60	C₂×Q₈	—	2	2	no	—
61	Q₁₆	—	2	2	no	—
62	Q₁₆	C₂×Dic₂₃₉	1	4	yes	—
63	C₂×C₈	—	239	1	no	—
64	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
65	C₄⋊C₄	—	956	2	no	—
66	C₄⋊C₄	—	478	2	no	—
67	C₄²	—	956	2	no	—
68	C₄²	—	239	1	no	—
69	C₁₆	—	478	1	no	—
70	C₁₆	C₂×Dic₂₃₉	1	2	yes	—
71	Q₈	—	2	4	no	—
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	956	2	no	—
75	C₂×C₄	—	239	1	no	—
76	C₂×C₄	—	239	1	no	—
77	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
78	C₈	—	478	1	no	—
79	C₈	C₂²×Dic₂₃₉	1	1	yes	characteristic
80	C₈	#16838547	1	1	yes	characteristic
81	C₂²	D₈×D₂₃₉	1	1	yes	characteristic
82	C₄	—	4	4	no	—
83	C₄	—	1912	2	no	—
84	C₄	—	478	1	no	—
85	C₄	—	239	2	no	—
86	C₄	D₄×Dic₂₃₉	1	1	yes	characteristic
87	C₄	#21439979	1	1	yes	characteristic
88	C₂	D₈×Dic₂₃₉	1	1	yes	characteristic
89	C₂	Q₃₂×D₂₃₉	1	1	yes	characteristic
90	C₂	#29505205	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₃₉
commutator	C₈×C₂₃₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	C₂×Q₃₂×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
cosocle	C₂×C₈×C₂₃₉	C₂³
hypercenter	C₂×Q₃₂	D₂₃₉
hypocenter	C₂₃₉	C₄×Q₃₂
LCS	G ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. Q₃₂×C₂₃₉	C₄	semidirect
41. Dic₂₃₉	Q₃₂	direct (N × H)
45. C₂₃₉	C₄×Q₃₂	semidirect
54. Q₃₂	Dic₂₃₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₁₆×Dic₂₃₉	2
#29504598	2
#29504576	2
C₁₆×Dic₂₃₉	2
C₂×Q₃₂×C₂₃₉	2
C₄×Q₃₂	239

Subgroups — Q32×Dic239 #44486407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₃₂×Dic₂₃₉ #44486407